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\documentclass{beamer}

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%\usepackage[utf8]{inputenc}
\usepackage{mystyle}
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\usetheme{Madrid}
\usecolortheme{beaver}
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%gets rid of bottom navigation bars
\setbeamertemplate{footline}[frame number]{}

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\title{Homology of strict $\omega$-categories}
%\subtitle{PhD defense}
\author{Léonard Guetta}
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\date{PhD defense, 28 January 2021}
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\institute{IRIF - Université de Paris}

\begin{document}

\frame{\titlepage}

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% \begin{frame}
%   \frametitle{Table of Contents}
%   \tableofcontents
% \end{frame}

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\begin{frame}
  \frametitle{Preliminary conventions}
  In this talk:
  \begin{itemize}
  \item<2-> $\oo$\nbd{}category = strict $\omega$\nbd{}category
    \item<3-> $n$\nbd{}category = $\oo$\nbd{}category with only unit cells above
      dimension $n$
      \item<4-> the functor $n\Cat \to \oo\Cat$ is an inclusion 
  \end{itemize}
  \end{frame}
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%%% oo-categories as spaces

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\begin{frame}
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  \frametitle{$\oo$\nbd{}categories as spaces}
  Starting point: Street's \emph{orientals}
  \[
    \Or \colon \Psh{\Delta} \to \oo\Cat.
  \]
  In pictures:
  \[
  \Or_0 = \bullet,
\]
\pause
  \[
    \Or_1=\begin{tikzcd}[ampersand replacement=\&]
     \bullet \ar[r] \&\bullet,
    \end{tikzcd}
  \]
  \pause
  \[
    \Or_2=
    \begin{tikzcd}[ampersand replacement=\&]
      \& \bullet \ar[rd]\& \\
    \bullet \ar[ru]\ar[rr,""{name=A,above}]\&\&\bullet,
    \ar[Rightarrow,from=A,to=1-2]
    \end{tikzcd}
  \]
  \pause
  \[
  \Or_3=
  \begin{tikzcd}[ampersand replacement=\&]
    \& \bullet \ar[rd]\& \\
    \bullet\ar[ru] \ar[rd,""{name=B,above}] \ar[rr,""{name=A,above}]\& \& \bullet \ar[ld]\\
    \& \bullet \&
    \ar[from=A,to=1-2,Rightarrow, shorten <= 0.25em, shorten >= 0.25em]
    \ar[from=B,to=2-3,Rightarrow, near start, shorten <= 1.1em, shorten >= 1.5em]
  \end{tikzcd}
 \Rrightarrow
    \begin{tikzcd}[ampersand replacement=\&]
    \& \bullet \ar[rd] \ar[dd,""{name=B,right}] \& \\
    \bullet\ar[ru] \ar[rd,""{name=A,above}] \& \& \bullet.  \ar[ld]\\
    \& \bullet \&
    \ar[from=A,to=1-2,Rightarrow, near start, shorten <= 1em, shorten >= 1.5em]
    \ar[from=B,to=2-3,Rightarrow, shorten <= 0.75em, shorten >=0.75em]
    \end{tikzcd}
    \]
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\end{frame}
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\begin{frame}
  \frametitle{$\oo$\nbd{}categories as spaces}
  \begin{block}{Definition}
    The \alert{nerve} of an $\oo$\nbd{}category $C$ is the simplicial
    set
    \[
  \begin{aligned}
        N_{\omega}(C) : \Delta^{\op} &\to \Set\\
      [n] &\mapsto \Hom_{\omega\Cat}(\Or_n,C).
    \end{aligned}
    \]
  \end{block}
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  \pause
  This yields the \alert{nerve functor} for $\oo$\nbd{}categories
  \[
    \begin{aligned}
      N_{\oo} : \oo\Cat &\to \Psh{\Delta} \\
      C &\mapsto N_{\oo}(C).
      \end{aligned}
  \]
\end{frame}
\begin{frame}
  \frametitle{$\oo$\nbd{}categories as spaces}
  TODO : Exemple en basse dimension
\end{frame}
\begin{frame}
  \frametitle{$\oo$\nbd{}categories as spaces}
  \begin{block}{Definition}
    A morphism $f\colon C \to D$ of $\oo\Cat$ is a \emph{Thomason equivalence}
    if $N_{\oo}(f)\colon N_{\oo}(C) \to N_{\oo}(D)$ is a weak equivalence of
    simplicial sets. 
  \end{block}
    \pause $\W^{\Th}$:=class of Thomason equivalences. \pause By definition, the
    nerve functor induces 
    \[
      \overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta}).
    \]
    Where:
    \begin{itemize}[label=$\bullet$]
      \item $\Ho(\oo\Cat^{\Th})$ is the localization of $\oo\Cat$ with respect to
        $\W^{\Th}$,
        \item $\Ho(\Psh{\Delta})$ is the localization of $\Psh{\Delta}$ with
          respect to weak equivalences of simplicial sets.
      \end{itemize}
    % Where $\Ho(-)$ stands for the localized category (or better
    % the localized pre-derivator or even weak $(\oo,1)$\nbd{}category).
  \end{frame}
\begin{frame}
  \frametitle{$\oo$\nbd{}categories as spaces}
  \begin{alertblock}{Theorem (Gagna, 2018)}
    $\overline{N_{\oo}} : \Ho(\oo\Cat^{\Th}) \to \Ho(\Psh{\Delta})$ is an
    equivalence of categories (or better an equivalence of derivators, or of weak $(\infty,1)$\nbd{}categories).
  \end{alertblock}
  \pause In other words:
  \begin{center}
    Homotopy theory of $\oo$\nbd{}categories induced by Thomason equivalences \\$\cong$\\ Homotopy theory of spaces
    \end{center}
  \end{frame}
  \begin{frame}
    \frametitle{Singular homology of $\oo$\nbd{}categories}
    Recall that we have the (normalized) chain complex functor
    \[
      \kappa \colon \Psh{\Delta} \to \Ch,
    \]
    Where $\Ch$ is the category of non-negatively graded chain complexes.\pause

    
    This functor sends weak equivalences of simplicial sets to quasi-isomorphisms.
    Hence,
    \[
      \overline{\kappa} \colon \Ho(\Psh{\Delta}) \to \Ho(\Ch),
    \]
    where $\Ho(\Ch)$ is the localization of $\Ch$ with respect to quasi-isomorphisms.
  \end{frame}
  \begin{frame}
    \frametitle{Singular homology of $\oo$\nbd{}categories}
    \begin{block}{Definition}
      The \emph{singular homology functor} $\sH^{\sing} \colon \Ho(\oo\Cat^{\Th}) \to
      \Ho(\Ch)$ is defined as the composition
      \[
        \Ho(\oo\Cat^{\Th}) \overset{\overline{N_{\oo}}}{\longrightarrow}
        \Ho(\Psh{\Delta}) \overset{\overline{\kappa}}{\longrightarrow} \Ho(\Ch).
      \]
    \end{block}
    \pause
    In pratice, this means that the $k$\nbd{}th singular homology group of an $\oo$\nbd{}category $C$ is
      the $k$\nbd{}th homology group of $N_{\oo}(C)$,
      \[
        H_k^{\sing}(C):=H_k(N_{\oo}(C)).
      \]
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  \end{frame}
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\end{document}
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